Chapter 4 | Applications of Derivatives
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CHAPTER 4 REVIEW
absolute extremum KEY TERMS
if f has an absolute maximum or absolute minimum at c , we say f has an absolute extremum
at c
if f ( c ) ≥ f ( x ) for all x in the domain of f , we say f has an absolute maximum at c if f ( c ) ≤ f ( x ) for all x in the domain of f , we say f has an absolute minimum at c a function F such that F ′( x ) = f ( x ) for all x in the domain of f is an antiderivative of f if f is differentiable over an interval I and f ′ is decreasing over I , then f is concave down over
absolute maximum absolute minimum
antiderivative concave down
I if f is differentiable over an interval I and f ′ is increasing over I , then f is concave up over I the upward or downward curve of the graph of a function suppose f is twice differentiable over an interval I ; if f ″>0 over I , then f is concave up over I ; if f ″<0 over I , then f is concave down over I if f ′( c ) =0 or f ′( c ) is undefined, we say that c is a critical point of f the differential dx is an independent variable that can be assigned any nonzero real number; the differential dy is defined to be dy = f ′( x ) dx given a differentiable function y = f ′( x ), the equation dy = f ′( x ) dx is the differential form of the derivative of y with respect to x the behavior of a function as x →∞ and x →−∞ if f is a continuous function over a finite, closed interval, then f has an absolute maximum and an absolute minimum if f has a local extremum at c , then c is a critical point of f let f be a continuous function over an interval I containing a critical point c such that f is differentiable over I except possibly at c ; if f ′ changes sign from positive to negative as x increases through c , then f has a local maximum at c ; if f ′ changes sign from negative to positive as x increases through c , then f has a local minimum at c ; if f ′ does not change sign as x increases through c , then f does not have a local extremum at c if lim x →∞ f ( x ) = L or lim x →−∞ f ( x ) = L , then y = L is a horizontal asymptote of f the most general antiderivative of f ( x ) is the indefinite integral of f ; we use the notation ∫ f ( x ) dx to denote the indefinite integral of f when evaluating a limit, the forms 0 0, ∞/∞, 0·∞, ∞−∞, 0 0 , ∞ 0 , and 1 ∞ are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is a function that becomes arbitrarily large as x becomes large if f is continuous at c and f changes concavity at c , the point ⎛ ⎝ c , f ( c ) ⎞ ⎠ is an inflection point of f
concave up concavity concavity test
critical point differential
differential form
end behavior extreme value theorem
Fermat’s theorem first derivative test
horizontal asymptote
indefinite integral
indeterminate forms
infinite limit at infinity inflection point
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